: Given any family of nonempty sets, their Cartesian product is a nonempty set.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given a set of integers, does some nonempty subset of them sum to 0?

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given a set

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Given the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.

Given a specific task to solve, and a class of functions, learning means using a set of observations to find which solves the task in some optimal sense.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given fixed resources, it is often better to spend more time collecting additional training data and more informative features than it is to spend extra time tuning the learning algorithms.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples into one category or the other.

Given some training data, a set of n points of the form

Given the rank of Lieutenant, he was posted to the Army School of Physical Training at Frankston, Victoria, to act as a divisional supervisor of physical training.

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other.

Given his later prowess, it is rather curious to note that during his martial training he apparently did not win many bouts.

Given a quarantine, he refused to quit boxing periodically and continued training during his time of illness.

Given the slight numerical and technological advantage of the Luftwaffe over its enemies in 1939-1941, its success during these years can largely be attributed to extensive officer and staff training programs along with the experiences of the Condor Legion in Spain.

Given his considerable martial ability, a result of hard and dedicated training, he was able to develop a unique and outstanding version of lion dance.

Given the expense of military pilot training, air forces typically conduct training in phases to winnow out unsuitable candidates.

Given that the nation was at peace, archery as well as other military martial arts became more of a method of personal development rather than military training.

Given the role of clergy in responding to crisis in families, Popenoe increased focus in training the clergy over many years.

Given these facts, waiting more than 48 hours between bouts of training for a particular muscle is a waste of time.

Given that UK specialist training takes 3 to 4 years longer than in other developed countries, it is evident that UK registrars continue to perform general medical duties in addition to their sub-specialty training.

Given a standard training set D of size n, bagging generates m new training sets, each of size n ′ < n, by sampling examples from D uniformly and with replacement.

Given a choice of missiles, and starting with a training mission, the player is sent after an enemy aircraft carrier, an enemy base and finally must destroy an enemy space shuttle.

Given his training in atmospheric physics, McDonald was able to examine UFO reports in greater detail than most other scientists, and was able to offer explanations for some previously unexplained reports.

Ironically, also nominated in the category of Contemporary Male Horse was Point Given who won under Baffert's training the Belmont Stakes, Preakness Stakes, Haskell Invitational Handicap, Travers Stakes and ultimately the Eclipse Award for Horse of the Year.

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

Given that there are many historical definitions of the keystone species concept, and without a consensus on its exact definition, a list of examples best illustrates the concept of keystone species.

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesized logic program which entails all the positive and none of the negative examples.

Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil ( based on working out some examples ) was striking and novel.

Given his high profile Laserna was absorbed into politics by those who wanted to present the world with one of the best examples of Colombian intellect, and so he served as the Colombian Ambassador to his native France ( 1976 – 1979 ) and to Austria ( 1987 – 1990 ), served in the Senate of Colombia and served as councillor of Bogotá.

Given that the programmes were pre-shot on film as opposed to being shown live, unlike most of the BBC's output from the late 1940s, examples of Television Newsreel do survive in the archives, some of the oldest extant pieces of BBC-produced television programming.